When utilizing a cluster decomposible relativistic scattering formalism, it is most convenient that the covariant field equations take on a linear form with respect to the energy and momentum dispersion on the fields in the manner given by the Dirac form for spin \half systems. A general spinor formulation is given for arbitrary spins by minimally extending the Lorentz algebra to include operators whose matrix representation give general Dirac matrices. The forms of these matrices are explicitly demonstrated for spin \half and spin 1 fields.
Using the complete orthonormal sets of radial parts of nonrelativitistic exponential type orbitals (2,1, 0, 1, 2, ...) and spinor type tensor spherical harmonics of rank s the new formulae for the 2(2s+1)-component relativistic spinors useful in the quantum mechanical description of the arbitrary half-integral spin particles by the generalized Dirac equation introduced by the author are established in position, momentum and four-dimensional spaces, where 1/ 2, 3 / 2, 5 / 2, ... s = . These spinors are complete without the inclusion of the continuum. The 2(2s+1)component spinors obtained are re…
This paper is the second one of a series of three and it is the continuation of math-ph/0412074. We review some properties of the algebraic spinors in Cl(3,0) and Cl(0,3) and how Weyl, Pauli and Dirac spinors are constructed in Cl(3,0) (and Cl(0,3) in the case of Weyl spinors. A plane wave solution for the Dirac equation is obtained, and the Dirac equation is written in terms of Weyl spinors, and alternatively, in terms of Pauli spinors. Finally the covariant and contravariant undotted spinors in Cl(0,3), isomorphic to two copies of the quaternion algebra (H), are constructed. We prove that th…
The three first sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself; it is based on an intrisic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibres over a real 4-dimensional manifold. The two following sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different persp…
We study the interplay between basic Dirac operator and transverse Killing and twistor spinors. In order to obtain results for general Riemannian foliations with bundle-like metric we consider transverse Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature it turns out that these Killing spinors and twistor spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, extending som…
The analytical relations in position, momentum and four-dimensional spaces are established for the expansion and one-range addition theorems of relativistic complete orthonormal sets of exponential type spinor wave functions and Slater spinor orbitals of arbitrary half-integral spin. These theorems are expressed through the corresponding nonrelativistic expansion and one-range addition theorems of the spin-0 particles introduced by the author. The expansion and one-range addition theorems derived are especially useful for the computation of multicenter integrals over exponential type spinor or…
Dirac linear spinor fields were obtained from non-linear Heisenberg spinors, in the literature. Here we extend that idea by considering not only Dirac spinor fields but spinor fields in any of the Lounesto's classes. When one starts considering all these classes of fields, the question of providing a classification for the Heisenberg spinor naturally arises. In this work the classification of Heisenberg spinor fields is derived and scrutinized, in its interplay with the Lounesto's spinor field classification.
Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong, together with Majorana spinor fields, to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class (5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. This paper is devoted to investigate and…
In this work, we consider a generalization of quantum electrodynamics including Lorentz violation and torsional-gravity, in the context of general spinor fields as classified in the Lounesto scheme. Singular spinor fields will be shown to be less sensitive to the Lorentz violation, as far as couplings between the spinor bilinear covariants and torsion are regarded. In addition, we prove that flagpole spinor fields do not admit minimal coupling to the torsion. In general, mass dimension four couplings are deeply affected when singular flagpole spinors are considered, instead of the usual Dirac …
In a previous paper we explicitly constructed a mapping that leads Dirac spinor fields to the dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields). ELKO spinor fields are prime candidates for describing dark matter, and belong to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification, based on the relations and values taken by their associated bilinear covariants. Such a mapping between Dirac and ELKO spinor fields was obtained in an attempt to extend the Standard Mode…
A classification of spinor fields according to the associated bilinear covariants is constructed in arbitrary dimensions and metric signatures, generalizing Lounesto's 4D spinor field classification. In such a generalized classification a basic role is played by the geometric Fierz identities. In 4D Minkowski spacetime the standard bilinear covariants can be either null or non-null -- with the exception of the current density which is invariably different from zero for physical reasons -- and sweep all types of spinor fields, including Dirac, Weyl, Majorana and more generally flagpoles, flag-d…
It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles - the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.
This paper deals with the classification of spinor fields according to the bilinear covariants in 7 dimensions. The previously investigated Riemannian case is characterized by either one spinor field class, in the real case of Majorana spinors, or three non-trivial classes in the most general complex case. In this paper we show that by imposing appropriate conditions on spinor fields in 7d manifolds with Lorentzian metric, the formerly obtained obstructions for new classes of spinor fields can be circumvented. New spinor fields classes are then explicitly constructed. In particular, on 7-manif…
Spinor fields depending on tensor fields and other spinor fields are considered. The concept of extended spinor fields is introduced and the theory of differentiation for such fields is developed.
We investigate the constraint equations of the Lounesto spinor fields classification and show that it can be used to completely characterize all the singular classes, which are potential accommodations for further mass dimension one fermions, beyond the well known Elko spinor fields. This result can be useful for two purposes: besides a great abridgement in the classification of a given spinor field, we provide a general form of each class of spinor fields, which can be used furthermore to search for a general classification of spinors dynamics.
Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators that transform gauged twistor spinors to gauged harmonic spinors are found. Symmetry operators of gauged harmonic spinors in terms of conformal Killing-Yano forms are obtained. Algebraic conditions to obtain solutions of the Seiberg-Witten equations are discussed.
Spinor representations of surfaces immersed into 4-dimensional pseudo-riemannian manifolds are defined in terms of minimal left ideals and tensor decompositions of Clifford algebras. The classification of spinor fields and Dirac operators on the immersed surfaces is given. The Dirac-Hestenes spinor field on surfaces immersed into Lorentzian manifolds and on surfaces conformally immersed into Minkowski spacetime is defined.
The Lounesto classification splits spinors in six classes: I, II, III are those for which at least one among scalar and pseudo-scalar bi-linear spinor quantities is non-zero, its spinors are called regular, and among them we find the usual Dirac spinor. IV, V, VI are those for which the scalar and pseudo-scalar bi-linear spinor quantities are identically zero, its spinors are called singular, and they are split in further sub-classes: IV has no further restrictions, its spinors are called flag-dipole; V is the one for which the spin axial-vector vanishes, its spinors are called flagpole, and a…
This work deals with new classes of spinors of mass dimension one in Minkowski spacetime. In order to accomplish it, the Lounesto classification scheme and the inversion theorem are going to be used. The algebraic framework shall be revisited by explicating the central point performed by the Fierz aggregate. Then the spinor classification is generalized in order to encompass the new mass dimension one spinors. The spinor operator is shown to play a prominent role to engender the new mass dimension one spinors, accordingly.
This paper proves that from the algebraic point of view ELKO spinor fields belong together with Majorana spinor fields to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class 5, according to Lounesto spinor field classification. The main algebraic difference between ELKO and Majorana spinor fields is that in the first case their respective two 2-component spinor fields present opposite helicities, while in the second case the helicity associated with the two 2-component of ELKO are not imposed to be flipped. The proof of our statement is based on Lou…
We consider gauged twistor spinors which are supersymmetry generators of supersymmetric and superconformal field theories in curved backgrounds. We show that the spinor bilinears of gauged twistor spinors satify the gauged conformal Killing-Yano equation. We prove that the symmetry operators of the gauged twistor spinor equation can be constructed from ordinary conformal Killing-Yano forms in constant curvature backgrounds. This provides a way to obtain gauged twistor spinors from ordinary twistor spinors.
Generalization of twistor spinors to Kähler manifolds which are called Kählerian twistor spinors are considered. We find the differential equation satisfied by the bilinear forms of Kählerian twistor spinors. We show that the bilinear form equation reduces to Kählerian conformal Killing-Yano equation under special conditions. We also investigate the special cases of holomorphic and anti-holomorphic Kählerian twistor spinors and the bilinear forms of alternative definitions for Kählerian twistor spinors.
We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor R matrices of low rank orthogonal algebras and the corresponding RTT algebras. Coincidences with fundamental R matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.
The space spinor formalism and estimates for spinor fields
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico IST, Universidade de Lisboa UL（里斯本大学高等技术研究所数学分析、几何与动力系统中心）；School of Mathematical Sciences, Queen Mary, University of London（伦敦大学玛丽女王学院数学科学学院）
The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing-Yano (KY) and conformal Killing-Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors respectively…
We consider geometric and supergravity Killing spinors and the spinor bilinears constructed out of them. The spinor bilinears of geometric Killing spinors correspond to the antisymmetric generalizations of Killing vector fields which are called Killing-Yano forms. They constitute a Lie superalgebra structure in constant curvature spacetimes. We show that the Dirac currents of geometric Killing spinors satisfy a Lie algebra structure up to a condition on 2-form spinor bilinears. We propose that the spinor bilinears of supergravity Killing spinors give way to different generalizations of Killing…
In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential operators on spinors which lead to the definitions of twistor and Killing spinors are discussed. Holonomy classification for manifolds admitting parallel and Killing spinors are given. Killing-Yano and conformal Killing-Yano forms resulting from the spinor bilinears of Killing and twistor spinors are introduced and the symmetry operators of special spinor e…
After reviewing the Lounesto spinor field classification, according to the bilinear covariants associated to a spinor field, we call attention and unravel some prominent features involving unexpected properties about spinor fields under such classification. In particular, we pithily focus on the new aspects --- as well as current concrete possibilities. They mainly arise when we deal with some non-standard spinor fields concerning, in particular, their applications in physics.
There are Poincare group representations on complex Hilbert spaces, like the Dirac spinor field, or real Hilbert spaces, like the electromagnetic field tensor. The Majorana spinor is an element of a 4 dimensional real vector space. The Majorana spinor field is a space-time dependent Majorana spinor, solution of the free Dirac equation. The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields are defined and related to the linear and angular momenta of a spin one-half representation of the Poincare group. We show that the Majorana spinor field with finite mass is an unitary…
Dual-helicity eigenspinors of the charge conjugation operator (ELKO spinor fields) belong -- together with Majorana spinor fields -- to a wider class of spinor fields, the so-called flagpole spinor fields, corresponding to the class-(5), according to Lounesto spinor field classification based on the relations and values taken by their associated bilinear covariants. There exists only six such disjoint classes: the first three corresponding to Dirac spinor fields, and the other three respectively corresponding to flagpole, flag-dipole and Weyl spinor fields. Using the mapping from ELKO spinor f…
Flagpole and flag-dipole spinors are particular classes of spinor fields that has been recently used in different branches of theoretical physics. In this paper, we study the possibility and consequences of these spinor fields to induce an underlying fluid flow structure in the background of Kerr spacetimes. We show that flag-dipole spinor fields are solutions of the equations of motion in this context. To our knowledge, this is the second time that this class of spinor field appears as a physical solution, the first one occurring as a solution of the Dirac equation in ESK gravities.
Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twis…
We present a supermatrix realisation of q-deformed spinor-spinor and spinor-vector R-matrices. These R-matrices are then used to construct transfer matrices for $U_{q^2}(\mathfrak{so}_{2n+1})$- and $U_{q}(\mathfrak{so}_{2n+2})$-symmetric closed spin chains. Their eigenvectors and eigenvalues are computed.
Two explicit formulas for metric connections in the bundle of Dirac spinors are studied. Their equivalence is proved. The explicit formula relating the spinor curvature tensor with the Riemann curvature tensor is rederived.
